CAREER: Research and Training at the Intersection of Number Theory and Analysis

素数几千年来一直是一个自然的研究对象，现在在数字通信的现代加密系统中发挥着基础作用。尽管他们进行了长期的研究，但关于质数分布的许多问题仍然没有解决。素数也与班数的研究交织在一起，班数出现在许多数论环境中。班级编号已经被研究了200年，尽管已经开发出了精确的猜想，但在很大程度上仍然是个谜。Radon变换在数学光谱上似乎很遥远，它量化了函数在低维表面上的“质量”分布；它们是支撑计算机断层扫描医学成像的理论的关键部分。Radon变换的研究是调和分析的中心领域，与Carleson算子和离散算术算子有着深远的联系，Carleson算子有助于回答关于傅立叶级数的一个历史问题，而离散算术算子则将调和分析与数论相结合。这个项目是数论和调和分析的交汇点，它探索并连接了所有这些主题。在工作过程中，该项目将通过博士后培训、暑期研究生班和儿童数学推广活动为数学界做出贡献。本研究的主要目标是数论和调和分析交叉点的五个项目。首先，得到了任意度数域的类数可除性的新界。第二，短特征标和将得到新的界，它在历史上为L函数提供了一个重要的次凸性结果，现在在多维环境中可能会影响涉及计算丢番图方程的积分解的问题。第三，在丢番图方程领域，将发展圆法的新变体，以及与多维振荡积分密切相关的问题。第四，将数论方法应用到调和分析的背景下，证明了离散算子的新结果。最后，对具有多项式相位和Radon型行为的Carleson算子进行了系统的研究。

Prime numbers have been a natural object of study for thousands of years and now play a foundational role in modern encryption systems for digital communications. Despite their long study, many questions about the distribution of prime numbers remain unsolved. Primes are also intertwined with the study of class numbers, which appear in many number-theoretic settings. Class numbers have been studied for 200 years, but still remain largely mysterious, although precise conjectures have been developed. Seemingly far away on the mathematical spectrum, Radon transforms quantify the distribution of the "mass" of functions along lower-dimensional surfaces; they are a critical part of the theory underlying Computed Tomography medical imaging. The study of Radon transforms is a central area in harmonic analysis with far-reaching connections both to the Carleson operator, which was instrumental in answering a historic question on Fourier series, and to the new world of discrete arithmetic operators, which blends harmonic analysis with number theory. This project, which is at the intersection of number theory and harmonic analysis, explores and connects all of these themes. During the course of the work, the project will contribute to the mathematical community through training postdocs, a graduate summer school, and mathematical outreach activities for children.The major aims of this research center on five projects at the intersection of number theory and harmonic analysis. First, new bounds relating to the divisibility of class numbers of number fields of arbitrary degree will be obtained. Second, new bounds will be obtained for short character sums, which historically provided an important subconvexity result for L-functions, and now in a multi-dimensional setting have the potential to impact problems involving counting integral solutions to Diophantine equations. Third, in the realm of Diophantine equations, new variations on the circle method, and closely related questions on multi-dimensional oscillatory integrals, will be developed. Fourth, new results for discrete operators will be proved by adapting number-theoretic methods to the setting of harmonic analysis. Finally, a systematic investigation of Carleson operators with polynomial phases and Radon-type behavior will be carried out.